Introduction to Linear Polynomials — Practice

Question 1

State the degree of 5x³ − 2x² + 7.

Accepted Answer

The highest power of x is 3.

✦ Answer: degree 3 (cubic).

Question 2

What is the coefficient of x² in 4x³ − 6x² + x − 9?

Accepted Answer

The term in x² is −6x².

✦ Answer: −6.

Question 3

Find the zero of the polynomial 2x + 6.

Accepted Answer

Step 1 — 2x + 6 = 0 → x = −6/2.

Step 2 — x = −3.

✦ Answer: x = −3.

Question 4

If p(x) = x² − 3x + 2, find p(2) and p(−1).

Accepted Answer

Step 1 — p(2) = 4 − 6 + 2 = 0.

Step 2 — p(−1) = 1 + 3 + 2 = 6.

✦ Answer: p(2) = 0, p(−1) = 6.

Question 5

Find the remainder when x³ + 3x² + 3x + 1 is divided by (x + 1).

Accepted Answer

Step 1 — divisor (x + 1) → evaluate at x = −1.

Step 2 — p(−1) = −1 + 3 − 3 + 1 = 0.

✦ Answer: remainder = 0.

Question 6

Show that (x − 1) is a factor of x³ − 1.

Accepted Answer

Step 1 — let p(x) = x³ − 1; test x = 1.

Step 2 — p(1) = 1 − 1 = 0.

Step 3 — since p(1) = 0, (x − 1) is a factor.

✦ Answer: p(1) = 0 ⇒ (x − 1) is a factor.

Question 7

Factorise x³ − 6x² + 11x − 6 using the Factor Theorem.

Accepted Answer

Step 1 — try x = 1: 1 − 6 + 11 − 6 = 0, so (x − 1) is a factor.

Step 2 — divide to get x² − 5x + 6.

Step 3 — factorise: x² − 5x + 6 = (x − 2)(x − 3).

✦ Answer: (x − 1)(x − 2)(x − 3).

Question 8

Find k if (x − 2) is a factor of x³ − kx² + 2x + 4.

Accepted Answer

Step 1 — (x − 2) is a factor → p(2) = 0.

Step 2 — p(2) = 8 − 4k + 4 + 4 = 16 − 4k = 0.

Step 3 — 4k = 16 → k = 4.

✦ Answer: k = 4.

Practice — Introduction to Linear Polynomials